Definition quadratic forms over a module

I have a question about the following definition from my book:

Originally Posted by definition

Let V be a module over a commutative ring A. A function Q: V -> A is called a quadratic form on V if the following properties are satisfied:
(1) $\displaystyle Q(ax) = a^2Q(x)$ for all $\displaystyle a\in A$ and $\displaystyle x\in V$
(2) $\displaystyle (x,y) -> Q(x+y) -Q(x)-Q(y)$ is a bilinear form.

My questions are the following:

When I looked up the definition of a quadratic form over a field property I saw that (1) is sufficient. Why should property (2) in this case also be included in the definition? Can anyone give an example of a function that is not a quadratic form, but for which (1) holds?